In
algebraic geometry a normal crossing singularity is a singularity similar to a union of coordinate hyperplanes. The term can be confusing because normal crossing singularities are not usually
normal schemes (in the sense of the local rings being integrally closed).
Normal crossing divisors
In
algebraic geometry, normal crossing divisors are a class of
divisors which generalize the smooth divisors. Intuitively they cross only in a transversal way.
The normal crossing points in the algebraic variety called the
Whitney umbrella are not simple normal crossings singularities.
The origin in the algebraic variety defined by is a simple normal crossings singularity. The variety itself, seen as a subvariety of the two-dimensional
affine plane is an example of a normal crossings divisor.
Any variety which is the union of smooth varieties which all have smooth intersections is a variety with normal crossing singularities. For example, let be irreducible polynomials defining smooth hypersurfaces such that the ideal defines a smooth curve. Then is a surface with normal crossing singularities.
References
Robert Lazarsfeld, Positivity in algebraic geometry, Springer-Verlag, Berlin, 1994.